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Simplifying (7x3 + -3x2 + 8x) + (8x3 + 6x2 + -13x) = 0 Reorder the terms: (8x + -3x2 + 7x3) + (8x3 + 6x2 + -13x) = 0 Remove parenthesis around (8x + -3x2 + 7x3) 8x + -3x2 + 7x3 + (8x3 + 6x2 + -13x) = 0 Reorder the terms: 8x + -3x2 + 7x3 + (-13x + 6x2 + 8x3) = 0 Remove parenthesis around (-13x + 6x2 + 8x3) 8x + -3x2 + 7x3 + -13x + 6x2 + 8x3 = 0 Reorder the terms: 8x + -13x + -3x2 + 6x2 + 7x3 + 8x3 = 0 Combine like terms: 8x + -13x = -5x -5x + -3x2 + 6x2 + 7x3 + 8x3 = 0 Combine like terms: -3x2 + 6x2 = 3x2 -5x + 3x2 + 7x3 + 8x3 = 0 Combine like terms: 7x3 + 8x3 = 15x3 -5x + 3x2 + 15x3 = 0 Solving -5x + 3x2 + 15x3 = 0 Solving for variable 'x'. Factor out the Greatest Common Factor (GCF), 'x'. x(-5 + 3x + 15x2) = 0Subproblem 1
Set the factor 'x' equal to zero and attempt to solve: Simplifying x = 0 Solving x = 0 Move all terms containing x to the left, all other terms to the right. Simplifying x = 0Subproblem 2
Set the factor '(-5 + 3x + 15x2)' equal to zero and attempt to solve: Simplifying -5 + 3x + 15x2 = 0 Solving -5 + 3x + 15x2 = 0 Begin completing the square. Divide all terms by 15 the coefficient of the squared term: Divide each side by '15'. -0.3333333333 + 0.2x + x2 = 0 Move the constant term to the right: Add '0.3333333333' to each side of the equation. -0.3333333333 + 0.2x + 0.3333333333 + x2 = 0 + 0.3333333333 Reorder the terms: -0.3333333333 + 0.3333333333 + 0.2x + x2 = 0 + 0.3333333333 Combine like terms: -0.3333333333 + 0.3333333333 = 0.0000000000 0.0000000000 + 0.2x + x2 = 0 + 0.3333333333 0.2x + x2 = 0 + 0.3333333333 Combine like terms: 0 + 0.3333333333 = 0.3333333333 0.2x + x2 = 0.3333333333 The x term is 0.2x. Take half its coefficient (0.1). Square it (0.01) and add it to both sides. Add '0.01' to each side of the equation. 0.2x + 0.01 + x2 = 0.3333333333 + 0.01 Reorder the terms: 0.01 + 0.2x + x2 = 0.3333333333 + 0.01 Combine like terms: 0.3333333333 + 0.01 = 0.3433333333 0.01 + 0.2x + x2 = 0.3433333333 Factor a perfect square on the left side: (x + 0.1)(x + 0.1) = 0.3433333333 Calculate the square root of the right side: 0.585946528 Break this problem into two subproblems by setting (x + 0.1) equal to 0.585946528 and -0.585946528.Subproblem 1
x + 0.1 = 0.585946528 Simplifying x + 0.1 = 0.585946528 Reorder the terms: 0.1 + x = 0.585946528 Solving 0.1 + x = 0.585946528 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.1' to each side of the equation. 0.1 + -0.1 + x = 0.585946528 + -0.1 Combine like terms: 0.1 + -0.1 = 0.0 0.0 + x = 0.585946528 + -0.1 x = 0.585946528 + -0.1 Combine like terms: 0.585946528 + -0.1 = 0.485946528 x = 0.485946528 Simplifying x = 0.485946528Subproblem 2
x + 0.1 = -0.585946528 Simplifying x + 0.1 = -0.585946528 Reorder the terms: 0.1 + x = -0.585946528 Solving 0.1 + x = -0.585946528 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.1' to each side of the equation. 0.1 + -0.1 + x = -0.585946528 + -0.1 Combine like terms: 0.1 + -0.1 = 0.0 0.0 + x = -0.585946528 + -0.1 x = -0.585946528 + -0.1 Combine like terms: -0.585946528 + -0.1 = -0.685946528 x = -0.685946528 Simplifying x = -0.685946528Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.485946528, -0.685946528}Solution
x = {0, 0.485946528, -0.685946528}
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